reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;

theorem Th33:
  a in S & S is Initial_Segm of T implies InitSegm(S,a) = InitSegm (T,a)
proof
  assume that
A1: a in S and
A2: S is Initial_Segm of T;
A3: S c= T by A2,Th29;
  thus InitSegm(S,a) c= InitSegm(T,a)
  proof
    let x be object;
    assume x in InitSegm(S,a);
    then x in LowerCone{a} & x in S by XBOOLE_0:def 4;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in InitSegm(T,a);
  then
A5: x in LowerCone{a} by XBOOLE_0:def 4;
  then consider a1 such that
A6: x = a1 and
A7: for a2 st a2 in {a} holds a1 < a2;
A8: a1 in T by A4,A6,XBOOLE_0:def 4;
  a in {a} by TARSKI:def 1;
  then a1 < a by A7;
  then x in S by A1,A2,A6,A8,Th32;
  hence thesis by A5,XBOOLE_0:def 4;
end;
